Essential Calculus: Early Transcendentals
2nd Edition
ISBN: 9781133112280
Author: James Stewart
Publisher: Cengage Learning
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Textbook Question
Chapter 12.4, Problem 18E
Find the moments of inertia Ix, Iy, I0 for the lamina of Exercise 12.
12. Find the center of mass of the lamina in Exercise 11 if the density at any point is proportional to the square of its distance from the origin.
11. A lamina occupies the part of the disk x2 + y2 ≤ 1 in the first quadrant. Find its center of mass if the density at any point is proportional to its distance from the x-axis.
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Chapter 12 Solutions
Essential Calculus: Early Transcendentals
Ch. 12.1 - (a) Estimate the volume of the solid that lies...Ch. 12.1 - If R = [0, 4] [1, 2], use a Riemann sum with m =...Ch. 12.1 - (a) Use a Riemann sum with m = n = 2 to estimate...Ch. 12.1 - (a) Estimate the volume of the solid that lies...Ch. 12.1 - A 20-ft-by-30-ft swimming pool is filled with...Ch. 12.1 - A contour map is shown for a function f on the...Ch. 12.1 - 79 Evaluate the double integral by first...Ch. 12.1 - 7-9 Evaluate the double integral by first...Ch. 12.1 - Evaluate the double integral by first identifying...Ch. 12.1 - The integral R9y2dA, where R = [0, 4] [0, 2],...
Ch. 12.1 - Calculate the iterated integral. 15....Ch. 12.1 - Calculate the iterated integral. 12....Ch. 12.1 - 1120 Calculate the iterated integral. 13....Ch. 12.1 - 1120 Calculate the iterated integral. 16....Ch. 12.1 - Calculate the iterated integral. 19....Ch. 12.1 - Calculate the iterated integral. 20. 1315lnyxydydxCh. 12.1 - Calculate the iterated integral. 21....Ch. 12.1 - Calculate the iterated integral. 24....Ch. 12.1 - Calculate the iterated integral. 25....Ch. 12.1 - Calculate the iterated integral. 26. 0101s+tdsdtCh. 12.1 - Calculate the double integral. 28....Ch. 12.1 - Calculate the double integral. 29....Ch. 12.1 - Calculate the double integral. 31....Ch. 12.1 - Prob. 26ECh. 12.1 - Calculate the double integral. 33....Ch. 12.1 - Calculate the double integral. 24....Ch. 12.1 - Sketch the solid whose volume is given by the...Ch. 12.1 - Sketch the solid whose volume is given by the...Ch. 12.1 - Find the volume of the solid that lies under the...Ch. 12.1 - Find the volume of the solid that lies under the...Ch. 12.1 - Find the volume of the solid lying under the...Ch. 12.1 - Find the volume of the solid enclosed by the...Ch. 12.1 - Find the volume of the solid enclosed by the...Ch. 12.1 - Find the volume of the solid in the first octant...Ch. 12.1 - Find the volume of the solid enclosed by the...Ch. 12.1 - Graph the solid that lies between the surface z =...Ch. 12.1 - Find the average value of f over the given...Ch. 12.1 - Find the average value of f over the given...Ch. 12.1 - If f is a constant function, f(x, y) = k, and R =...Ch. 12.1 - Use the result of Exercise 41 to show that...Ch. 12.1 - Use symmetry to evaluate the double integral. 49....Ch. 12.1 - Use symmetry to evaluate the double integral. 50....Ch. 12.1 - Prob. 46ECh. 12.2 - 16 Evaluate the iterated integral. 1. 040yxy2dxdyCh. 12.2 - Evaluate the iterated integral. 2. 012x2(xy)dydxCh. 12.2 - 16 Evaluate the iterated integral. 3....Ch. 12.2 - Evaluate the iterated integral. 2. 02y2yxydxdyCh. 12.2 - Evaluate the iterated integral. 5....Ch. 12.2 - Evaluate the iterated integral. 6. 010ex1+exdwdvCh. 12.2 - 710 Evaluate the double integral. 7....Ch. 12.2 - Evaluate the double integral. 8....Ch. 12.2 - 710 Evaluate the double integral. 9....Ch. 12.2 - Evaluate the double integral. 10....Ch. 12.2 - Express D as a region of type I and also as a...Ch. 12.2 - Express D as a region of type I and also as a...Ch. 12.2 - Set up iterated integrals for both orders of...Ch. 12.2 - Set up iterated integrals for both orders of...Ch. 12.2 - Evaluate the double integral. 17.DxcosydA, D is...Ch. 12.2 - Evaluate the double integral. 18. D(x2+2y)dA, D is...Ch. 12.2 - Evaluate the double integral. 19. Dy2dA, D is the...Ch. 12.2 - Evaluate the double integral. 18....Ch. 12.2 - Prob. 19ECh. 12.2 - 1520 Evaluate the double integral. 20. D2xydA, D...Ch. 12.2 - 2130 Find the volume of the given solid. 21. Under...Ch. 12.2 - Prob. 22ECh. 12.2 - Prob. 23ECh. 12.2 - Prob. 24ECh. 12.2 - 2130 Find the volume of the given solid. 25....Ch. 12.2 - Find the volume of the given solid. 28. Bounded by...Ch. 12.2 - Find the volume of the given solid. 29. Enclosed...Ch. 12.2 - Find the volume of the given solid. 30. Bounded by...Ch. 12.2 - Find the volume of the given solid. 31. Bounded by...Ch. 12.2 - Prob. 30ECh. 12.2 - Prob. 31ECh. 12.2 - Prob. 32ECh. 12.2 - Sketch the solid whose volume is given by the...Ch. 12.2 - Sketch the solid whose volume is given by the...Ch. 12.2 - Sketch the region of integration and change the...Ch. 12.2 - Sketch the region of integration and change the...Ch. 12.2 - Sketch the region of integration and change the...Ch. 12.2 - Sketch the region of integration and change the...Ch. 12.2 - Sketch the region of integration and change the...Ch. 12.2 - Prob. 42ECh. 12.2 - Evaluate the integral by reversing the order of...Ch. 12.2 - 43-48 Evaluate the integral by reversing the order...Ch. 12.2 - 4348 Evaluate the integral by reversing the order...Ch. 12.2 - Prob. 46ECh. 12.2 - Evaluate the integral by reversing the order of...Ch. 12.2 - Evaluate the integral by reversing the order of...Ch. 12.2 - Express D as a union of regions of type I or type...Ch. 12.2 - Express D as a union of regions of type I or type...Ch. 12.2 - 5152 Use Property 11 to estimate the value of the...Ch. 12.2 - Use Property 11 to estimate the value of the...Ch. 12.2 - Prove Property 11.Ch. 12.2 - In evaluating a double integral over a region D, a...Ch. 12.2 - Prob. 55ECh. 12.2 - Prob. 56ECh. 12.2 - Prob. 57ECh. 12.2 - Prob. 58ECh. 12.2 - Prob. 59ECh. 12.3 - 14 A region R is shown. Decide whether to use...Ch. 12.3 - Prob. 2ECh. 12.3 - Prob. 3ECh. 12.3 - Prob. 4ECh. 12.3 - Sketch the region whose area is given by the...Ch. 12.3 - Prob. 6ECh. 12.3 - Evaluate the given integral by changing to polar...Ch. 12.3 - Prob. 8ECh. 12.3 - Evaluate the given integral by changing to polar...Ch. 12.3 - Prob. 10ECh. 12.3 - Prob. 12ECh. 12.3 - Prob. 11ECh. 12.3 - Use a double integral to find the area of the...Ch. 12.3 - Use a double integral to find the area of the...Ch. 12.3 - Prob. 13ECh. 12.3 - Prob. 14ECh. 12.3 - Use polar coordinates to find the volume of the...Ch. 12.3 - Prob. 15ECh. 12.3 - Use polar coordinates to find the volume of the...Ch. 12.3 - 1319 Use polar coordinates to find the volume of...Ch. 12.3 - Use polar coordinates to find the volume of the...Ch. 12.3 - (a) A cylindrical drill with radius r1 is used to...Ch. 12.3 - 2326 Evaluate the iterated integral by converting...Ch. 12.3 - Evaluate the iterated integral by converting to...Ch. 12.3 - 2326 Evaluate the iterated integral by converting...Ch. 12.3 - Evaluate the iterated integral by converting to...Ch. 12.3 - A swimming pool is circular with a 40-ft diameter....Ch. 12.3 - An agricultural sprinkler distributes water in a...Ch. 12.3 - Use polar coordinates to combine the sum...Ch. 12.3 - (a) We define the improper integral (over the...Ch. 12.3 - Use the result of Exercise 30 part (c) to evaluate...Ch. 12.4 - Electric charge is distributed over the rectangle...Ch. 12.4 - Electric charge is distributed over the disk x2 +...Ch. 12.4 - Find the mass and center of mass of the lamina...Ch. 12.4 - Find the mass and center of mass of the lamina...Ch. 12.4 - Find the mass and center of mass of the lamina...Ch. 12.4 - 3-10 Find the mass and center of mass of the...Ch. 12.4 - Find the mass and center of mass of the lamina...Ch. 12.4 - 3-10 Find the mass and center of mass of the...Ch. 12.4 - 310 Find the mass and center of mass of the lamina...Ch. 12.4 - 3-10 Find the mass and center of mass of the...Ch. 12.4 - A lamina occupies the part of the disk x2 + y2 1...Ch. 12.4 - Find the center of mass of the lamina in Exercise...Ch. 12.4 - The boundary of a lamina consists of the...Ch. 12.4 - Find the center of mass of the lamina in Exercise...Ch. 12.4 - Find the center of mass of a lamina in the shape...Ch. 12.4 - A lamina occupies the region inside the circle x2...Ch. 12.4 - Find the moments of inertia Ix, Iy, I0 for the...Ch. 12.4 - Find the moments of inertia Ix, Iy, I0 for the...Ch. 12.4 - Find the moments of inertia Ix, Iy, lo for the...Ch. 12.4 - Consider a square fan blade with sides of length 2...Ch. 12.4 - A lamina with constant density (x, y) = occupies...Ch. 12.4 - A lamina with constant density (x, y) = occupies...Ch. 12.5 - Evaluate the integral in Example 1, integrating...Ch. 12.5 - Evaluate the integral E(xy+z2)dv, where...Ch. 12.5 - Evaluate the iterated integral....Ch. 12.5 - 36 Evaluate the iterated integral. 5....Ch. 12.5 - 00x0xzx2sinydydzdxCh. 12.5 - Evaluate the iterated integral. 6....Ch. 12.5 - Evaluate the triple integral. 9. EydV, where...Ch. 12.5 - Evaluate the triple integral. 10.EezydV, where...Ch. 12.5 - Evaluate the triple integral. 11. Ezx2+z2dV, where...Ch. 12.5 - Evaluate the triple integral. 12. EsinydV, where E...Ch. 12.5 - Evaluate the triple integral. 13. E6xydV, where E...Ch. 12.5 - Prob. 12ECh. 12.5 - 716 Evaluate the triple integral. 13. T x2 dV,...Ch. 12.5 - 7-16 Evaluate the triple integral. 14. TxyzdV,...Ch. 12.5 - Evaluate the triple integral. 17. ExdV, where E is...Ch. 12.5 - Evaluate the triple integral. 18. EzdV, where E is...Ch. 12.5 - Prob. 17ECh. 12.5 - Use a triple integral to find the volume of the...Ch. 12.5 - Use a triple integral to find the volume of the...Ch. 12.5 - Use a triple integral to find the volume of the...Ch. 12.5 - Prob. 23ECh. 12.5 - Prob. 24ECh. 12.5 - Prob. 25ECh. 12.5 - Prob. 26ECh. 12.5 - Express the integralEf(x,y,z)dV, as an iterated...Ch. 12.5 - Prob. 28ECh. 12.5 - Prob. 29ECh. 12.5 - Prob. 30ECh. 12.5 - Prob. 31ECh. 12.5 - Prob. 32ECh. 12.5 - Write five other iterated integrals that are equal...Ch. 12.5 - Prob. 34ECh. 12.5 - Prob. 35ECh. 12.5 - Prob. 36ECh. 12.5 - 3740 Find the mass and center of mass of the solid...Ch. 12.5 - Prob. 38ECh. 12.5 - Prob. 39ECh. 12.5 - Prob. 40ECh. 12.5 - Prob. 45ECh. 12.5 - Prob. 46ECh. 12.5 - Prob. 47ECh. 12.5 - Prob. 48ECh. 12.5 - Prob. 41ECh. 12.5 - Prob. 42ECh. 12.5 - Prob. 44ECh. 12.5 - Prob. 49ECh. 12.5 - Prob. 50ECh. 12.6 - Plot the point whose cylindrical coordinates are...Ch. 12.6 - Prob. 2ECh. 12.6 - Prob. 3ECh. 12.6 - Prob. 4ECh. 12.6 - Prob. 5ECh. 12.6 - Prob. 6ECh. 12.6 - 78 Identify the surface whose equation is given....Ch. 12.6 - Prob. 8ECh. 12.6 - Prob. 9ECh. 12.6 - Prob. 10ECh. 12.6 - Prob. 11ECh. 12.6 - Prob. 12ECh. 12.6 - Prob. 13ECh. 12.6 - Prob. 14ECh. 12.6 - Sketch the solid whose volume is given by the...Ch. 12.6 - Sketch the solid whose volume is given by the...Ch. 12.6 - Use cylindrical coordinates. 17. Evaluate...Ch. 12.6 - Prob. 18ECh. 12.6 - Prob. 19ECh. 12.6 - 21-32 Use spherical coordinates. 20. Evaluate...Ch. 12.6 - Use cylindrical coordinates. 21. Evaluate Ex2dV,...Ch. 12.6 - Prob. 22ECh. 12.6 - Use cylindrical coordinates. 23. Find the volume...Ch. 12.6 - Prob. 24ECh. 12.6 - 1728 Use cylindrical coordinates. 25. (a) Find the...Ch. 12.6 - Use cylindrical coordinates. 26. (a) Find the...Ch. 12.6 - Use cylindrical coordinates. 27. Find the mass and...Ch. 12.6 - Use cylindrical coordinates. 28. Find the mass of...Ch. 12.6 - Evaluate the integral by changing to cylindrical...Ch. 12.6 - Prob. 30ECh. 12.6 - Prob. 31ECh. 12.7 - Prob. 1ECh. 12.7 - Prob. 2ECh. 12.7 - Prob. 3ECh. 12.7 - Prob. 4ECh. 12.7 - Prob. 5ECh. 12.7 - Prob. 6ECh. 12.7 - 78 Identify the surface whose equation is given....Ch. 12.7 - Identify the surface whose equation is given. 8. ...Ch. 12.7 - Prob. 9ECh. 12.7 - Prob. 10ECh. 12.7 - 1114 Sketch the solid described by the given...Ch. 12.7 - Sketch the solid described by the given...Ch. 12.7 - 1112 Sketch the solid described by the given...Ch. 12.7 - Sketch the solid described by the given...Ch. 12.7 - A solid lies above the cone z = x2+y2 and below...Ch. 12.7 - Prob. 16ECh. 12.7 - Prob. 17ECh. 12.7 - Sketch the solid whose volume is given by the...Ch. 12.7 - Prob. 19ECh. 12.7 - Prob. 20ECh. 12.7 - Use spherical coordinates. 21. Evaluate B (x2+y2 +...Ch. 12.7 - 21-32 Use spherical coordinates. 22. Evaluate...Ch. 12.7 - Prob. 23ECh. 12.7 - 21-32 Use spherical coordinates. 24. Evaluate...Ch. 12.7 - Use spherical coordinates. 25. Evaluate E xe x2 +...Ch. 12.7 - Prob. 26ECh. 12.7 - Use spherical coordinates. 29. (a) Find the volume...Ch. 12.7 - Use spherical coordinates. 30. Find the volume of...Ch. 12.7 - Prob. 29ECh. 12.7 - Use spherical coordinates. 32. Let H be a solid...Ch. 12.7 - Prob. 31ECh. 12.7 - Use spherical coordinates. 34. Find the mass and...Ch. 12.7 - Use cylindrical or spherical coordinates,...Ch. 12.7 - Use cylindrical or spherical coordinates,...Ch. 12.7 - Evaluate the integral by changing to spherical...Ch. 12.7 - Evaluate the integral by changing to spherical...Ch. 12.7 - Evaluate the integral by changing to spherical...Ch. 12.7 - A model for the density of the earths atmosphere...Ch. 12.7 - Use a graphing device to draw a silo consisting of...Ch. 12.7 - Prob. 42ECh. 12.7 - Show that x2+y2+z2e-(x2+y2+z2) dx dy dz = 2 (The...Ch. 12.7 - Prob. 45ECh. 12.8 - 16 Find the Jacobian of the transformation. 1. x =...Ch. 12.8 - Find the Jacobian of the transformation. 2. x =...Ch. 12.8 - 16 Find the Jacobian of the transformation. 3. x =...Ch. 12.8 - Find the Jacobian of the transformation. 4. x =...Ch. 12.8 - 16 Find the Jacobian of the transformation. 5. x =...Ch. 12.8 - Find the Jacobian of the transformation. 6. x = v...Ch. 12.8 - Find the image of the set S under the given...Ch. 12.8 - Find the image of the set S under the given...Ch. 12.8 - Find the image of the set S under the given...Ch. 12.8 - Find the image of the set S under the given...Ch. 12.8 - A region R in the xy-plane is given. Find...Ch. 12.8 - Prob. 12ECh. 12.8 - A region R in the xy-plane is given. Find...Ch. 12.8 - A region R in the xy-plane is given. Find...Ch. 12.8 - Use the given transformation to evaluate the...Ch. 12.8 - Use the given transformation to evaluate the...Ch. 12.8 - Use the given transformation to evaluate the...Ch. 12.8 - Use the given transformation to evaluate the...Ch. 12.8 - Use the given transformation to evaluate the...Ch. 12.8 - Use the given transformation to evaluate the...Ch. 12.8 - (a) Evaluate E dV, where E is the solid enclosed...Ch. 12.8 - An important problem in thermodynamics is to find...Ch. 12.8 - Evaluate the integral by making an appropriate...Ch. 12.8 - Evaluate the integral by making an appropriate...Ch. 12.8 - Evaluate the integral by making an appropriate...Ch. 12.8 - Evaluate the integral by making an appropriate...Ch. 12.8 - Evaluate the integral by making an appropriate...Ch. 12.8 - Let f be continuous oil [0, 1] and letRbe the...Ch. 12 - Prob. 1RCCCh. 12 - Prob. 2RCCCh. 12 - Prob. 3RCCCh. 12 - Prob. 4RCCCh. 12 - Prob. 7RCCCh. 12 - Prob. 5RCCCh. 12 - Suppose a solid object occupies the region E and...Ch. 12 - Prob. 8RCCCh. 12 - (a) If a transformation T is given by x = g(u, v),...Ch. 12 - Determine whether the statement is true or false....Ch. 12 - Determine whether the statement is true or false....Ch. 12 - Determine whether the statement is true or false....Ch. 12 - Determine whether the statement is true or false....Ch. 12 - Determine whether the statement is true or false....Ch. 12 - Determine whether the statement is true or false....Ch. 12 - Determine whether the statement is true or false....Ch. 12 - Determine whether the statement is true or false....Ch. 12 - Determine whether the statement is true or false....Ch. 12 - A contour map is shown for a function f on the...Ch. 12 - Use the Midpoint Rule to estimate the integral in...Ch. 12 - Calculate the iterated integral. 3....Ch. 12 - Calculate the iterated integral. 4. 0101yexydxdyCh. 12 - Calculate the iterated integral. 5....Ch. 12 - Calculate the iterated integral. 6. 01xex3xy2dydxCh. 12 - Calculate the iterated integral. 7....Ch. 12 - Calculate the iterated integral. 8....Ch. 12 - Write Rf(x,y)dA as an iterated integral, where R...Ch. 12 - Write Rf(x,y)dA as an iterated integral, where R...Ch. 12 - Prob. 39RECh. 12 - Prob. 40RECh. 12 - Prob. 41RECh. 12 - Prob. 42RECh. 12 - Prob. 43RECh. 12 - Prob. 44RECh. 12 - Describe the region whose area is given by the...Ch. 12 - Prob. 12RECh. 12 - Prob. 13RECh. 12 - Prob. 14RECh. 12 - Prob. 15RECh. 12 - Prob. 16RECh. 12 - Prob. 17RECh. 12 - Prob. 18RECh. 12 - Prob. 19RECh. 12 - Prob. 20RECh. 12 - Prob. 21RECh. 12 - Prob. 22RECh. 12 - Prob. 23RECh. 12 - Prob. 24RECh. 12 - Prob. 25RECh. 12 - Prob. 26RECh. 12 - Prob. 27RECh. 12 - Prob. 28RECh. 12 - Prob. 29RECh. 12 - Prob. 30RECh. 12 - Prob. 31RECh. 12 - Prob. 32RECh. 12 - Prob. 33RECh. 12 - Prob. 34RECh. 12 - Prob. 35RECh. 12 - Prob. 36RECh. 12 - Prob. 37RECh. 12 - Use polar coordinates to evaluate...Ch. 12 - Use spherical coordinates to evaluate...Ch. 12 - Rewrite the integral 11x2101yf(x,y,z)dzdydxas an...Ch. 12 - Prob. 48RECh. 12 - Use the transformation u = x y, v = x + y to...Ch. 12 - Use the transformation x = u2, y = v2 z = w2 to...Ch. 12 - Use the change of variables formula and an...Ch. 12 - Prob. 52RE
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